ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Ext zero as Hom in both constructions

Example

Write the degree-zero identifications in the projective and injective complexes and compare them through balanced Ext.

Facts & Assumptions

Given: Objects $M, N$ in an abelian category, an augmented projective resolution $P_{∙} ↠ M$ , and an augmented injective resolution $N ↪ I^{∙}$ .

Verification

technique · direct

1.1givenalgebra

The augmentation identifies $M = coker (d_{1} : P_{1} \to P_{0})$ , so left exactness of $Hom (-, N)$ identifies $\ker (Hom (P_{0}, N) \to Hom (P_{1}, N))$ with $Hom (M, N)$ . Thus ${Ext}_{P}^{0} (M, N) = Hom (M, N)$ .

2.1step 1.1algebra∎

Likewise $N = \ker (d^{0} : I^{0} \to I^{1})$ , and left exactness of $Hom (M, -)$ gives $\ker (Hom (M, I^{0}) \to Hom (M, I^{1})) = Hom (M, N)$ . The two identifications are the degree-zero maps used in balanced ${Ext}^{0} (M, N)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 (standard reference, not scraped)